Spanish Math 68
Dog Years Dilemma
Trina was playing with her new puppy last night. She began to think about what she had read in a book about dogs. It said that for every year a dog lives it actually is the same as 7 human years. She looked at her 41/2 monthold puppy and wondered how many human years old her puppy was?
Using as much math language and good reasoning as you can, figure out how many human years old Trina's puppy is?
Suggested Grade Span
Grades 68
Alternative Versions of the Task
More Accessible Version:
Trina was playing with her puppy last night. She began to think about what she had read in a book about dogs. It said that for every year a dog lives it actually is the same as 7 human years. She looked at her 4yearold puppy and wondered how many human years old her puppy was?
Using as much math language and good reasoning as you can, figure out how many human years old Trina's puppy is?
More Challenging Version:
Trina was playing with her new puppy last night. She began to think about what she had read in a book about dogs. It said that for every year a dog lives it actually is the same as 7 human years. She liked at her 41/2 monthold puppy and wondered how many human years old her puppy was?
Using as much math language and good reasoning as you can, figure out how many human years old Trina's puppy is?
Newer research has found that the simple formula of multiplying a dog’s age by 7 to get its human age equivalent may not be that simple. Scientists have concluded that the weight of a pet is also a factor that needs to be considered when calculating the age. Analyze the chart below to write formulas for determining a dog’s human age equivalent for each of the 4 weight categories.
Pet's age 
020 Lbs  21 – 50 lbs  5190 lbs  Greater than 90 lbs 
5  36  37  40  42 
6  40  42  45  49 
7  44  47  50  56 
8  48  52  55  63 
9  52  57  60  70 
10  56  62  65  77 
11  60  67  70  84 
12  64  72  75  91 
13  68  77  80  98 
14  72  82  85  105 
15  76  87  90  
16  80  92  95  
17  84  97  100  
18  88  102  105  
19  92  107  110  
20  96  112  115 
Context
Trina, one of my sixthgrade students, called me one night all excited because she knew she had a math "dilemma" for the class. I was pleased that she recognized that a fair amount of mathematics was necessary to solve this problem and that it would be challenging for the class to solve. I actually was surprised at how difficult the problem was for my students. We were working on adding and subtracting fractions, so this problem dealing with fractions was quite timely. My students had enough mathematics to solve the problem, but used many different strategies because it was a problem that they had never encountered before.
What This Task Accomplishes
At first, the task reads fairly simple. However, many students could not think of a way to get started, so this task makes students persevere. It also makes them try different approaches as they begin sorting out the problem. It makes many students begin converting fractions to decimals, so they can work with a calculator.
What Students Will Do
Some students wanted to figure out the equivalent in dog age of one human day. They knew that months had different days, but they figured they would be very close. Some that got really frustrated rounded 41/2 months to 6 months and found that to be 31/2 yearsold in dog age. This was reasonable, but I encouraged them to get a more accurate answer. Because we do a lot of problem solving with charts, some kids made a chart and found that very successful for this problem, although it might not have been as successful with other fractional parts of a year.
Time Required for Task
Approximately 60 minutes. It took 45 minutes to solve the problem and another 15 minutes to pull their answers together and report.
Interdisciplinary Links
This problem can lead very nicely to a discussion on different life spans and conjectures about why some animals live longer than others.
Teaching Tips
I let my students work in pairs to solve this problem. You might ask your students to think about how graphing could lead them to a solution.
NCTM Standards
 Numbers and Operations
 Geometry and Measurement
Concepts to be Assessed and Skills to be Developed
 Multiplication
 Division
 Fractions/Decimals
 Functions
 Time/Scheduling
 Ratios
 Computation with decimals
 Relationships
Suggested Materials
 Paper
 Pencil
 Calculators
Possible Solutions
Orignal Version:
(4.5 months/12) x 7 years = 31.5/12 years = 2.625 years = 2 5/8 years
More Accessible Version:
7 x 4 = 28 years old
More Challenging Version:
x = Dog years y = Human years
020 lbs: 4x + 16 = y
2150 lbs: 5x + 17 = y
5190 lbs: 5x = 15 = y
Greater than 90 lbs: 7x + 7= y
TaskSpecific Assessment Notes
Task Specific Rubric/Benchmark Descriptors Click on a level for student example. 


Novice  This student has set up the facts that s/he knows about the calendar and is trying different algorithms to solve the problem with little or no reasoning. There is no evidence of a strategy and no explanation of the reasons for the algorithms tried. There is no mathematical representation. 
Apprentice  This student uses a strategy that would work, but uses faulty reasoning in changing a decimal fraction of a year to months. S/he correctly divides to find the dog age equivalent to one human month. However, s/he incorrectly assumes that .6 of a year is six months. The rest of the solution is based on that faulty reasoning. 
Practitioner  This student's strategy shows s/he has an understanding of the problem and the major concepts necessary for a solution. Their chart shows equivalent dog years for fractions of human years. This strategy of taking half of each human year leads to a solution of this problem (it may not have gotten a solution to other age puppies). There is effective mathematical reasoning. The student sees that halving the human age would also halve the dog years. The explanation is clear and the chart is appropriate use of mathematical representation. The student also uses correct mathematical notation. 
Expert  This student shows a deep understanding of the problem including the ability to identify the appropriate mathematical concepts. This student realizes that no matter how old the dog is, you need to multiply the age by seven. S/he realizes that the age of the dog is 4.5/12 of a year. Since s/he is unfamiliar with multiplying fractions, s/he used his/her knowledge of the fraction line as division and found the decimal equivalent of 4.5/12. This is a very efficient and sophisticated strategy that employs refined and complex reasoning. There is a clear and effective explanation and the student reached for a generalization that would solve any month old dog. The graph also actively communicates how to estimate the dog age of any living dog. 
Novice
Apprentice
Practitioner
Expert
Novice
Apprentice
Practitioner
Expert